The following is left unproven in a monograph. The author refers to it as "elementary exercise" but I am unable to prove it. Any insight is appreciated. $$ \int f \log f d\mu \le 2 \left[\int|f-1|^p d\mu\right]^{1/p}+\frac{2}{p-1}\int |f-1|^p d\mu,\quad p>1 $$ where $f$ is a probability density.
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Minor formatting and Math Jaxing: I added nothing to the question, so if there's one of the reviewers with more than 2000 experience points, please reject my edit and then edit it again in order to improve the formatting of this question
Daniele Tampieri
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An "elementary" inequality
Daniel Li
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